Cap vs Bruce would be a hard fight, but in the end Cap would take it. And I'll explain how. There have been many direct comparisons of the two combatants in this thread already, and most of them have been right, with the exception of a few.

Batman is a member of his world's premier hero team, is their master tactician, and takes on godlike superbeings with his brains, skills, and brilliant prepwork.

Captain America is the leader of his world's premier hero team, is their master tactician, and takes on godlike superbeings with his brains, skills...and nothing else.

But the major factor here is this: Experience.

They have been published since about the same time, but Batman's progression has been linear, he was born, he ages, he fights crime. He took a little time jaunt recently but it was relatively short.

Captain America has been active since World War 2, but he spent decades frozen. But the deciding factor here is Kang. Kang the Conqueror is a time-traveling villain who really hates the Avengers. In one storyline, he kidnaps Cap and takes him to a world Kang already rules with an iron fist, and tells Cap to try and usurp him. Cap creates a rebellion, makes plans, leads them against Kang...and fails. Kang's forces are to strong. So, Kang takes Cap back again for another try, but lets him keep his memories of the failed attempt. Cap tries new tactics, doesn't win. This process continues for a hundred years of relative time, from Cap's point of view. He doesn't age, Kang keeps him young, but he keeps all the experiences. Like an extremely violent version of Groundhog Day. After a hundred years of constant fighting (from Cap's perspective) Kang takes him home, admitting he could not break Cap's spirit. That is the level of experience Batman is up against. That is why he cannot win.